Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras

Benjamin Dequêne, Sunny Roy

TL;DR

The paper develops a framework to study exact structures on hereditary abelian categories, introducing the diamond exact structure and Gen-Sub operators to generate well-behaved subcategories. It connects these subcategories to tilting theory via $ ext{E}$-mutations and shows that maximal canonically Jordan recoverable subcategories in type $A$ quivers are precisely the subcategories $ ext{GS}_{ ext{E}_{ riangle}}(T)$ generated by tilting objects under $ ext{E}_{ riangle}$-mutations. It provides combinatorial and algebraic characterizations of canonical Jordan recoverability in type $A_n$ via adjacency-avoiding interval sets and ties these to Gen-Sub closures of tilting classes. The work also develops a lattice-theoretic perspective, proving that the $ oughly_{ ext{E}_{ riangle}}$ relation induces a lattice congruence on the tilting lattice, with conjectured Boolean quotients in general, and outlines extensions via cokernel–kernel operators and further directions for exploring exact-structure mutations and recoverability phenomena.

Abstract

On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver $(Q,R)$, a full additive subcategory of $\operatorname{rep}(Q,R)$ is said to be Jordan recoverable if any $X \in \mathscr{C}$ can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory $\mathscr{C}$ is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back $X \in \mathscr{C}$ from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators $\operatorname{GS}_\mathcal{E}$, parametrized by the exact structures $\mathcal{E}$ of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over type $A$ quivers endowed with the diamond exact structure $\mathcal{E}_\diamond$, we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form $\operatorname{GS}_{\mathcal{E}_\diamond}(T)$ for some tilting object $T$.

Exact structures and maximal canonically Jordan recoverable subcategories for modules over type $A$ algebras

TL;DR

The paper develops a framework to study exact structures on hereditary abelian categories, introducing the diamond exact structure and Gen-Sub operators to generate well-behaved subcategories. It connects these subcategories to tilting theory via -mutations and shows that maximal canonically Jordan recoverable subcategories in type quivers are precisely the subcategories generated by tilting objects under -mutations. It provides combinatorial and algebraic characterizations of canonical Jordan recoverability in type via adjacency-avoiding interval sets and ties these to Gen-Sub closures of tilting classes. The work also develops a lattice-theoretic perspective, proving that the relation induces a lattice congruence on the tilting lattice, with conjectured Boolean quotients in general, and outlines extensions via cokernel–kernel operators and further directions for exploring exact-structure mutations and recoverability phenomena.

Abstract

On one hand, exact structures were introduced by D. Quillen in the '70s. They can be defined as collections of short exact sequences in a fixed abelian category satisfying additional properties. On the other hand, in a recent work, A. Garver, R. Patrias, and H. Thomas introduced Jordan recoverability. Given a bounded quiver , a full additive subcategory of is said to be Jordan recoverable if any can be recovered, up to isomorphism, from the Jordan form of its generic nilpotent endomorphisms. Such a subcategory is said to be canonically Jordan recoverable if, moreover, there exists a precise algebraic procedure that allows one to get back from that same Jordan form data. We introduce a new family of operators, called Gen-Sub operators , parametrized by the exact structures of abelian categories. After showing some properties of those operators in hereditary abelian categories, by focusing on the setting of modules over type quivers endowed with the diamond exact structure , we establish that the maximal canonically Jordan recoverable subcategories are precisely of the form for some tilting object .

Paper Structure

This paper contains 25 sections, 60 theorems, 17 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Exact structures on hereditary abelian categories
  3. Canonical Jordan recoverability
  4. Main results
  5. Outline of the paper
  6. Quiver representations
  7. Vocabulary
  8. Representations
  9. ADE Dynkin type quivers
  10. Gen-Sub operators on exact categories
  11. Exact structures
  12. Gen-Sub operators
  13. Adaptation to an exact structure preserving
  14. Extension closure preserving
  15. Tilting objects in hereditary subcategories
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $Q$ be an $A_n$ type quiver for some $n \in \mathbb{N}^*$. A subcategory $\mathscr{C} \subseteq \operatorname{rep}(Q)$ is canonically Jordan recoverable if and only if for any nonsplit short exact sequence \begin{tikzcd} 0 & E & F & G & 0, \arrow[from=1-1, to=1-2] \arrow[tail, from=1-2, to=

Figures (9)

  • Figure 1: Calculation of $\operatorname{GS}_{\mathcal{E}_\diamond}(\mathscr{C})$ where $\mathscr{C} = \operatorname{add} \left( \right)$. We get $\operatorname{GS}_{\mathcal{E}_\diamond}^1(\mathscr{C}) = \operatorname{add} \left( + \right)$ and $\operatorname{GS}_{\mathcal{E}_\diamond}^2(\mathscr{C}) = \operatorname{add} \left( + + \right)$. Here, we obtain that $\operatorname{GS}_{\mathcal{E}_\diamond}(\mathscr{C}) = \operatorname{GS}_{\mathcal{E}_\diamond}^2(\mathscr{C})$.
  • Figure 2: The Auslander--Reiten quiver of $Q$ in \ref{['ex:Eadaptednonextclosed']}. We have $\mathscr{C} = \operatorname{add} \left( \right) = \operatorname{GS}_{\mathcal{E}_{\diamond}}(\mathscr{C})$, and we can check that $\mathscr{C}$ is not closed under extensions even though it is $\mathcal{E}_{\diamond}$-adapted.
  • Figure 3: Tilting representations of $\operatorname{rep}(Q)$ for some $A_4$ type quiver $Q$. A tilting object in this figure is the direct sum of indecomposable objects in for one copy of the Auslander--Reiten quiver of $Q$
  • Figure 4: The Auslander--Reiten quiver of $Q$ in \ref{['ex:Eadaptednonextclosed']}. We have $\mathscr{C} = \operatorname{add} \left( \right) = \operatorname{GS}_{\mathcal{E}'}(\mathscr{C})$, and we can check that $\mathscr{C}$ is not closed under extensions even though it is a maximal $\mathcal{E}'$-adapted subcategory of $\mathscr{A}$.
  • Figure 5: Example of a maximal canonically Jordan recoverable subcategory $\mathscr{C} = \operatorname{add} \left( \right)$ in $\operatorname{rep}(Q)$, for $Q$ given in \ref{['fig:GSCalc1']}.
  • ...and 4 more figures

Theorems & Definitions (135)

  • Theorem 1.1: D23
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: G72
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 125 more